## _wp06229 — Reserves as a Share of GDP in Asia and Latin America, 1980-2003

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---

### Introduction: purpose and main findings
- Objective:
  - Calibrate a model of the optimal level of reserves for emerging market countries as insurance against “sudden stops” in capital flows associated with falls in output.
- Mechanism:
  - Reserves allow the government to smooth domestic absorption in crises; cost is that reserves yield a lower return than the interest rate the government must pay on long-term external liabilities.
- Analytical contribution:
  - Closed-form formula derived for the optimal level of reserves and calibrated using data on a sample of emerging market countries.
- Key quantitative finding:
  - For a coefficient of constant relative risk aversion of 2, the model predicts a reserves-to-GDP ratio of 10 percent, which is “very close” to the average reserves-to-GDP ratio observed in a group of 34 middle-income countries over the period 1980-2003.
- Performance and limits:
  - Model can account for a significant part of cross-regional variation in observed reserves, but does not account for the recent large increase in reserves in emerging market Asian countries.
- Output:
  - Provides estimates of the share of observed reserves that can be justified as insurance against sudden stops and a decomposition between the insurance-justified component and the residual component (conditional on assumed risk aversion).

### Sudden stops and reserves: stylized facts (empirical evidence)
- Sudden stop identification:
  - sudden stop in year t ⇔ k_t − k_{t−1} < −5%.
- Sample and data:
  - 34 middle-income countries over 1975-2003 (countries classified as middle-income by the World Bank, plus Korea; excludes major oil-producing countries).
  - Financial account, change in reserves, income and transfers from IMF International Financial Statistics; converted to constant local currency using nominal exchange rate vis-à-vis U.S. dollar and local GDP deflator; real GDP and real GDP deflator from World Bank World Development Indicators.
  - Exceptional financing and IMF loans counted as reserves rather than capital inflows.
- Average behavior in event window (events after 1980; exclude sudden stops inside five-year window of previous sudden stop):
  - Large fall in the financial account in the year of the sudden stop, amounting to more than 9.5 percent of the previous year’s output.
  - Most of the negative impact of the financial account reversal is offset by a fall in reserve accumulation.
  - Domestic absorption falls by only 2.6 percent of GDP on average in the year of the sudden stop.
  - Real growth falls at the time of the sudden stop but remains positive.
- Counterfactual:
  - If reserves accumulation were the same in the year of the sudden stop as in the year before, domestic absorption would fall by 8.7 percent of output on average instead of 2.6 percent.
- Illustrative episode — Uruguay 2002:
  - Capital account reversal amounted to 38 percentage points of GDP.
  - Government used a large amount of foreign exchange reserves (a significant part available in the context of an IMF arrangement).
  - Decline in domestic absorption was 15 percent of GDP—much smaller than the shock to the capital account.

### The model: setup, assumptions, and instruments
- Time and goods:
  - Discrete time t = 0,1,2,...
  - Single good consumed domestically and abroad.
  - Deterministic path disturbed by an exogenous sudden stop (only source of uncertainty).
- Agents:
  - Representative consumer with CRRA preferences; government issues a long-term security and accumulates reserves.
- Sudden stop modeling:
  - Sudden stop: fall in output by fraction γ below long-run growth path; private external debt L falls to zero in a sudden stop (normalization).
  - Probability of sudden stop each period: π. After sudden stop all uncertainty resolved and economy grows at rate g_r < g.
  - Private external debt share pre-sudden-stop: λ (level of private external debt as share of output in pre-sudden-stop period).
- Government instruments and costs:
  - Government long-term security price P satisfies P = 1/(r + δ + π?) and implied expression P = 1/(r + δ + π) (text provides step forms).
  - δ is a term premium; price constant before sudden stop and falls to zero when sudden stop occurs.
  - Reserves relation: R_t = P_t N_t (stock of reserves financed by number N_t of securities).
  - Assumption: 1δπ+< so transfer is positive. After sudden stop government becomes inactive: R_t, N_t, Z_t = 0.
- Private consumption before, during, after sudden stop summarized by equations (11)–(13) in the source.

### Optimal reserves: first-order condition and closed-form formula
- Government objective:
  - Maximize U = Σ_{s=0,...,∞} β^s u(C_{t+s}) with u(C) = (C^{1−σ} − 1)/(1 − σ).
  - Period-t choice maximizes expected utility of period t+1 consumption: arg max_{R_t} [π u(C_{t+1}^d) + (1 − π) u(C_{t+1}^b)].
- First-order condition (equation (15)) and marginal rate of substitution:
  - p_t ≡ u′(C^d_t)/u′(C^b_t); under optimality p is constant with closed-form expression involving π and δ (text expression).
  - If δ = 0 then p = 1 (perfect insurance). If δ > 0 then p > 1 (consumption lower in sudden stop).
- Closed-form optimal reserves-to-output ratio (equation (16) in source):
  - ρ = [ r g^(1/σ) / (p^(1/σ) g^(1/σ) + ... ) ] (exact algebra preserved in source; full expression in paper).
- Approximate formula (equation (17), obtained by setting 0rg δπ+=− = in last term):
  - ρ ≈ (1/p)^{1/σ} (1 − ρλγ)?? (text gives approximate expression: 1/(1) .p σ ρλγ − ≈+−− — see source for exact symbolic form).
- Intuition from approximate formula:
  - Optimal reserves increase one-for-one with amount of short-term debt λ and output cost of sudden stop γ.
  - If δ = 0 then p = 1 and reserves should fully smooth the sudden stop: ρ = λ γ + (text indicates full smoothing).
  - Optimal reserves fall with an increase in p (caused by decrease in π or increase in δ).
  - Increase in risk aversion σ decreases 1/p^{σ} and raises optimal reserves.
- Relation to Greenspan-Guidotti rule:
  - Greenspan-Guidotti: reserves-to-short-term-debt ratio should equal 1 (ρ λ = 1).
  - Greenspan-Guidotti perfectly smoothes consumption in a sudden stop if there is no output cost.
  - Model: optimal reserves could be higher (to smooth output loss γ) or lower (because of opportunity cost captured by δ and π effects).

### Calibration and benchmark economy
- Parameters and calibration targets:
  - Seven parameters determine behavior: probability of sudden stop π, output loss ratio γ, ratio of short-term debt to GDP λ, return on reserves r, term premium δ, risk-aversion σ (text enumerates parameters).
- Empirical calibration choices:
  - Unconditional probability of sudden stop in sample: 10.2 percent per year, rounded to 0.1 (π = 0.1) in calibration.
  - Parameter λ calibrated as average level of short-term debt; available numeric value: λ close to 11 percent (λ = 0.11 in Table 2).
  - If π = 0.1, life expectancy of the long-term security is ten years (life expectancy = 1/π).
- Benchmark values (from Table 2 and narrative):
  - Size of sudden stop λ = 0.11 [0,0.3]
  - Probability of a sudden stop π = 0.10 [0,0.25]
  - Output loss γ = 0.065 [0,0.2]
  - Potential output growth g = 0.033
  - Term premium δ = 0.015 [0.0025,0.05]
  - Risk free rate r = 0.05
  - Risk aversion σ = 2 [1,10]
- Benchmark outcomes:
  - Optimal level of reserves = 10.1 percent of GDP, or 92 percent of the short-term external debt.
  - Observed average reserves-to-GDP during 1975-2003 = 9.4 percent.

### Sensitivity analysis and comparative rules
- Greenspan-Guidotti comparison:
  - Greenspan-Guidotti provides a good approximation for how optimal reserves vary with the size of sudden stops.
  - Optimal ratio of reserves to short-term debt remains in the 90 to 100 percent range if the size of the sudden stop exceeds 10 percent of GDP.
  - Optimal level of reserves is zero if short-term debt amounts to less than 2.5 percent of GDP.
- Sensitivity to key parameters (exact values from source):
  - Doubling π from 5 percent to 10 percent increases optimal reserves from 4.6 percent to 10.1 percent of GDP.
  - Increasing δ from 1.5 percent to 3 percent reduces optimal reserve-to-GDP ratio from 10.2 percent to 3.8 percent.
  - Risk aversion effects:
    - Increasing σ from 1 to 4 raises optimal reserves from 3 percent to 14 percent of GDP.
    - Increasing σ from 6 to 8 has a much milder impact due to strong concavity.

### Estimating sudden-stop risk and impacts of fundamentals
- Probit estimation (34 middle-income countries, 1975-2003):
  - Preferred specification (regression 3.1) finds π decreases with pre-crisis growth and increases with currency real appreciation, public debt-to-GDP, financial openness (absolute gross inflows as a share of GDP), and ratio of foreign liabilities to money in the banking sector.
  - Fixed exchange rate dummy associated with higher π only if financial openness and exchange rate overvaluation are omitted.
- Size of sudden stops:
  - Take average size of sudden stop = 10.8 percent of GDP (λ ≈ 0.108; Table 2 uses λ = 0.11).
- Benchmark predicted probability and reserves:
  - Probit model (regression 3.1) predicts π = 7.9 percent for the average economy, implying optimal reserves-to-GDP = 8.6 percent.
- Impacts of changes in fundamentals (Table 5 exact entries summarized):
  - 20 percent real appreciation → estimated annual π rises by approximately 4.3 percentage points; optimal reserves increase by 2.7 percentage points of GDP.
  - Public debt/GDP rising from 40 percent to 60 percent → optimal reserve ratio increases by 1.7 percentage points of GDP.
  - Foreign liabilities to money (Thailand example: 45 → 262) → optimal reserves raise by 3.7 percentage points of GDP.
  - One standard deviation increase in financial openness → optimal reserves increase by 3.2 percentage points of GDP.
  - Change from floating to fixed exchange rate regime → optimal reserves increase by 3.4 percentage points of GDP.
- Initial optimal level reported in Table 5:
  - Initial optimal level of reserves: equivalent to 8.2 percent of GDP (about 3 months of imports, or 105 percent of short-term debt).

### Regional trends and model performance
- Aggregation approach:
  - Optimal reserves computed for each country-year for 34 middle-income countries; π from probit estimates; size of sudden stop set to realized mean per region and decade; σ set to 6 to match mean reserves-to-GDP in Asia in 1991.
- Asia:
  - Model suggests reserves should have declined between early and late 1980s due to slowdown in financial flows after early 1980s debt crisis.
  - Beginning early 1990s, model implies rapid increase in optimal reserves owing to rising international financial integration.
  - Slight decline in optimal reserves after Asian crises mainly due to reductions in public debt and financial flows.
  - Model outperforms three-months-of-imports and Greenspan-Guidotti rules in predicting actual reserve levels for most of the period considered.
  - Post-1997 buildup: model suggests buildup in reserves has been excessive. To justify observed accumulation between 1997 and 2003, expected size of either the sudden stop or the output cost would have had to more than double relative to its 1990s average. Increasing σ from 6 to 10 would raise warranted reserves by less than 3 percentage points of GDP—insufficient to explain the post-crisis buildup.
- Latin America:
  - Model suggests relatively high optimal reserves in 1980s, lower in early 1990s (improved fundamentals), and rising during the past decade with heightened financial integration.
  - Close match between model and actual data in 1991-2003 despite no individual-year Latin America reserves being used in calibration.
  - On average, current reserves may be adequate, though some large individual countries have insufficient reserve coverage.

### Extensions and robustness checks (selected)
- A. Crisis prevention (endogenous π):
  - Variant assumes π decreases with pre-crisis reserves Π(ρ) with Π′ < 0.
  - If sudden stop can be triggered by sunspot when short-term debt not fully covered by reserves:
    - Π(ρ) = sπ for ρ < λρs; Π(ρ) = 0 for ρ ≥ λρs (notation in text).
  - For benchmark calibration with sπ = 11 percent, Greenspan-Guidotti rule is optimal: increasing reserves from 92 percent to 100 percent of short-term debt is worth the cost.
  - Endogenizing π could increase or decrease optimal reserves in general.
- B. Real exchange rate depreciation:
  - If external liabilities and reserves are in foreign currency and real exchange rate depreciates by ΔQ at sudden stop, value of reserves in domestic consumption rises.
  - Net effect ambiguous; benchmark: a real depreciation of 10 percent increases optimal reserves from 10.1 percent to 10.2 percent of GDP (negligible).
- C. Endogenizing private-sector behavior:
  - Representative consumer faces borrowing constraint b_t ≤ λY_t^b before sudden stop.
  - Binding condition reformulated; condition satisfied for benchmark calibration, so private behavior consistent with intertemporal optimization.
  - Government reserves reduce private incentives to accumulate reserves; restricting private short-term debt issuance would be welfare-decreasing in model.
- D. Multiple sudden stops:
  - Episodes can last θ periods; profile determined by exogenous parameters {γ_s, λ_s} for s = 0,...,θ. Section III corresponds to special case θ = +∞ with specified profiles leading to earlier formula validity.

### Concluding comments — substantive findings and research directions
- Core contribution:
  - Derives a simple formula for the optimal level of international reserves based on the assumption that the main benefit of reserves is to smooth domestic absorption against disruptions induced by sudden stops in capital flows.
- Three directions for future work:
  1. Role of default risk in the opportunity cost of holding reserves — unclear whether default risk premium should be counted like the term premium; default may itself be a smoothing device.
  2. Preventive role of reserves in reducing probability and severity of sudden stops — requires empirical estimation of two-way causality to calibrate extended model.
  3. Collective management of reserves — benefits of reserve pooling and consequences of an institution (e.g., IMF) that relaxes external credit constraints invite multicountry extensions.

*Source: _wp06229 (IMF Working Paper excerpt — “Reserves as a Share of GDP in Asia and Latin America, 1980-2003”).*

### References..............................................................................................................

### _wp06229 - References..........................................................................................................................32

### Tables
- 1.        Countries and Years of Sudden Stops.....................................................................22
- 2.        Calibration Parameters............................................................................................23
- 3.        Probit Estimation of the Probability of a Sudden Stop ...........................................23
- 4.        Variables Used in Regression Analysis ..................................................................24
- 5.        Changes in Fundamentals and Optimal Reserves ...................................................25

### Figures
- 1.        International Reserves as a Share of GDP ..............................................................26
- 2.        Domestic Absorption and International Reserves in Sudden Stops, 1980-2003 ....27
- 3.        Domestic Absorption and International Reserves in Sudden Stops in the Model ..28
- 4.        The Optimal Level of Reserves: Sensitivity Analysis ............................................29

*Source: _wp06229 - References..............................................................................................................*

### 5.        Reserves as a Share of GDP in Asia and Latin America, 1980-2003.....................30

### 5.        Reserves as a Share of GDP in Asia and Latin America, 1980-2003

### Introduction: purpose and main findings
- Objective: calibrate a model of the optimal level of reserves for emerging market countries as insurance against “sudden stops” in capital flows associated with falls in output.
- Main mechanism: reserves allow the government to smooth domestic absorption in crises; cost is that reserves yield a lower return than the interest rate the government must pay on long-term external liabilities.
- Closed-form formula derived for the optimal level of reserves and calibrated using data on a sample of emerging market countries.
- Key quantitative finding: for a coefficient of constant relative risk aversion of 2, the model predicts a reserves-to-GDP ratio of 10 percent, which is “very close” to the average reserves-to-GDP ratio observed in a group of 34 middle-income countries over the period 1980-2003.
- Model performance: can account for a significant part of cross-regional variation in observed reserves, but does not account for the recent large increase in reserves in emerging market Asian countries.
- Contribution: provides estimates of the share of observed reserves that can be justified as insurance against sudden stops and a decomposition between the insurance-justified component and the residual component (conditional on assumed risk aversion).

### Sudden stops and reserves: stylized facts (empirical evidence)
- Sudden stop identification rule (Guidotti and others (2004) criterion used): a sudden stop in year t if the ratio of capital inflows to GDP, k_t ≡ K_t/Y_t, falls by more than 5 percent of GDP relative to the previous year: 1
  sudden stop in year t ⇔ k_t − k_{t−1} < −5%.
- Sample: 34 middle-income countries over 1975-2003 (countries classified as middle-income by the World Bank, plus Korea; excludes major oil-producing countries).
- Data sources and transforms: financial account, change in reserves, income and transfers from IMF International Financial Statistics; converted from current U.S. dollar to constant local currency units using nominal exchange rate vis-à-vis U.S. dollar and local GDP deflator; real GDP and real GDP deflator from World Bank World Development Indicators. Exceptional financing and IMF loans are counted as reserves rather than capital inflows.
- Average behavior in event window centered on sudden stop (events after 1980; exclude sudden stops that occur inside five-year window of previous sudden stop):
  - Large fall in the financial account in the year of the sudden stop, amounting to more than 9.5 percent of the previous year’s output.
  - Most of the negative impact of the financial account reversal is offset by a fall in reserve accumulation.
  - Domestic absorption falls by only 2.6 percent of GDP on average in the year of the sudden stop—much less than the financial account reversal.
  - Real output contribution is relatively small: real growth falls at the time of the sudden stop but remains positive.
- Counterfactual: if reserves accumulation were the same in the year of the sudden stop as in the year before, domestic absorption would fall by 8.7 percent of output on average instead of 2.6 percent.
- Illustrative episode: Uruguay in 2002
  - Capital account reversal amounted to 38 percentage points of GDP.
  - Government used a large amount of foreign exchange reserves (a significant part available in the context of an IMF arrangement).
  - Decline in domestic absorption was 15 percent of GDP—much smaller than the shock to the capital account.

### The model: setup and assumptions
- Time and goods:
  - Discrete time t = 0,1,2,...
  - Single good consumed domestically and abroad.
  - Deterministic path disturbed by an exogenous sudden stop (only source of uncertainty).
- Agents:
  - Representative consumer (private sector) with budget constraint (equation (4)):
    1
    (1),
    ttttt
    CYLrL  Z
    −
    =+−+   +
  - Government issues a long-term security and accumulates reserves.
- Sudden stop modeling:
  - Sudden stop: debt rollover crisis associated with a fall in output by fraction γ below its long-run growth path; representative consumer unable to roll over external debt; private external debt L falls to zero in a sudden stop (normalization).
  - After the sudden stop private external debt remains zero; output returns to long-run growth path.
  - Sudden stop probability each period: π. After sudden stop all uncertainty resolved and economy grows at rate g_r < g.
  - Notation: superscripts b, d, a denote periods before, during, and after sudden stop.
  - Private external debt share pre-sudden-stop: λ (level of private external debt as share of output in pre-sudden-stop period).
  - Dynamics summarized in equations (5) and (6):
    00
    (1),     (1) (1),
    batdt
    ttt
    YYgYYgYγ==+=− +
    0
    (1),     0,
    btda
    ttt
    LgYLLλ=+==
- Government instruments and costs:
  - Government long-term security: pays one unit of good every period until sudden stop; price P satisfies P = 1/(r + δ + π?) — text gives formula steps: []
    1
    1(1   )   ,
    1
    PP
    r
    π
    δ
    =+− ⋅
    ++
    implying,
    1
    .P
    r
    δπ
    =
    ++
  - Interpretation: δ is a term premium; price constant before sudden stop and falls to zero when sudden stop occurs; interest rate used to compute present value of long-term security is higher than short-term rate r.
  - Reserves relation (equation (7)): R_t = P_t N_t (stock of reserves financed by number N_t of securities).
  - Government budget constraint (equation (8)) and transfer before sudden stop (equation (9)):
    11
    1
    ().
    b
    ttt
    ZrRR
    P
    δπ
    −−
    ⎛⎞
    =−   −=−  +
    ⎜⎟
    ⎝⎠
  - Transfer in sudden stop (equation (10)):
    1
    (1).
    d
    tt
    ZRδπ
    −
    =−−
  - Assumption: 1δπ+< so transfer is positive. After sudden stop government becomes inactive: R_t, N_t, Z_t = 0.
- Private consumption expressions before, during, after sudden stop (equations (11)-(13)):
  - Before (equation (11)):
    11
    (1)(),
    bbbb
    tt ttt
    CYLrLRδπ
    −−
    =+−+  −+
  - During (equation (12)):
    11
    (1)(1)(1),
    dbb
    tttt
    CYrLRγδπ
    −−
    =−  −+   +−−
  - After (equation (13)):
    .
    aa
    tt
    CY=
- Two interpretations/remarks:
  - Accumulating reserves is equivalent to lengthening maturity of consolidated external debt (substituting short-term debt L by long-term debt PN). Aggregated constraint shown in text.
  - Model takes private behavior as given; assumption that government enjoys comparative advantage in issuing long-term securities can make private sector behavior consistent with intertemporal optimization (addressed in Section V).

### Optimal reserves: first-order condition and closed-form formula
- Government objective: maximize representative consumer welfare U = Σ_{s=0,...,∞} β^s u(C_{t+s}) with CRRA utility u(C) = (C^{1−σ} − 1)/(1 − σ) (equation (14)).
- Period-t choice maximizes expected utility of period t+1 consumption:
  arg max_{R_t} (1) π u(C_{t+1}^d) + (1 − π) u(C_{t+1}^b)
- First-order condition (equation (15)):
  ()()
  11
  (1)(1) ().
  db
  tt
  uCuCπδππδπ
  ++
  ′′
  −−=−  +
- Define marginal rate of substitution p_t between consumption in sudden-stop and non-sudden-stop states:
  ()
  ()
  .
  d
  t
  t
  b
  t
  uC
  p
  uC
  ′
  ≡
  ′
- Under optimality p is constant and equal to (text expression):
  1
  1.
  1(1)
  p
  πδπδ
  πδπ   π δπ
  −+
  ≡=+
  −−−−
- If δ = 0 then p = 1 (perfect insurance, C^d = C^b). If δ > 0 then p > 1 (consumption lower in sudden stop).
- Closed-form optimal reserves-to-output ratio (equation (16)):
  1/
  1/
  1
  1()().
  1(1)(1)1
  prg
  pg
  σ
  σ
  ρλγλ δπλγ
  δπ
  ⎛⎞
  −−
  =+−−    − +  +
  ⎜⎟
  +−−−  +
  ⎝⎠
- Approximate formula for parameter ranges considered (equation (17)), obtained by setting 0rg
  δπ+=− = in last term:
  1/
  (1) .p
  σ
  ρλγ
  −
  ≈+−−
- Intuition from approximate formula:
  - Optimal reserves increase one-for-one with amount of short-term debt λ and output cost of sudden stop γ.
  - If δ = 0 then p = 1 and reserves should fully smooth the sudden stop: ρ = λ γ + (text uses ρλγ=+).
  - Optimal reserves fall with an increase in p, which can be caused by a decrease in π or an increase in δ.
  - Increase in risk aversion σ decreases 1/p^{σ} and raises optimal reserves.
- Relation to Greenspan-Guidotti rule:
  - Greenspan-Guidotti: reserves-to-short-term-debt ratio should equal 1, i.e., ρ λ = (text states) .ρλ=
  - Greenspan-Guidotti perfectly smoothes consumption in a sudden stop if there is no output cost.
  - Our model: optimal reserves could be higher (to smooth output loss γ) or lower (because of cost of holding reserves captured by δ and π effects).

### Applications: calibration approach (overview)
- Calibration targets and parameters:
  - Seven parameters determine behavior: probability of sudden stop π, output loss ratio γ, ratio of short-term debt to GDP λ, return on reserves r, term premium δ, risk-aversion parameter σ (text states seven but lists six here in narrative; model discussion enumerates parameters).
- Empirical calibration choices (as described up to available text):
  - Three parameters (π, λ, γ) calibrated using sample of sudden stops.
  - Unconditional probability of sudden stop in sample: 10.2 percent per year, rounded to 0.1π= in calibration.
  - Parameter λ calibrated as average level of (k_t − k_{t−1})? (text: parameter λ was calibrated as the average level of 1() t t k k − − over sample of sudden stops, which is close to 11 percent). The sentence in source cuts off mid-explanation; available numeric value: λ close to 11 percent.
- Example interpretation of π and life expectancy:
  - If 0.1π= (a plausible value, as noted), the life expectancy of the long-term security is ten years (life expectancy = 1/π).

*Source: IMF Working Paper excerpt — “Reserves as a Share of GDP in Asia and Latin America, 1980-2003” (excerpted pages provided).*

### 8.2 percent on average in our sample according to the World Bank’s Global Development

### _wp06229 - 8.2 percent on average in our sample according to the World Bank’s Global Development

### Calibration and benchmark economy
- Output cost of a sudden stop:
  - Growth rate falls by 4.5 percent on average relative to normal times in a sudden stop and by 2.2 percent in the following year.
  - Benchmark γ set to 6.5 percent.
- Interest rates and growth:
  - Risk-free short-term dollar interest rate r = 5 percent.
  - Term premium, 1.5δ = percent, equals the average differential between the yield on 10-year U.S. Treasury bonds and the federal fund rate during 1987-2005.
  - Growth rate g = 3.3 percent (average real GDP growth rate in sample of middle-income countries during 1975-2002, excluding sudden-stop years).
- Risk aversion:
  - Benchmark risk aversion and its range of variation follow standard growth and real business cycle literature.
- Benchmark optimal reserves:
  - Optimal level of reserves = 10.1 percent of GDP, or 92 percent of the short-term external debt.
  - Observed average reserves-to-GDP during 1975-2003 = 9.4 percent.

### Sensitivity analysis and comparative rules
- Greenspan-Guidotti rule comparison:
  - Greenspan-Guidotti rule provides a good approximation for how optimal reserves vary with the size of sudden stops.
  - Optimal ratio of reserves to short-term debt remains in the 90 to 100 percent range if the size of the sudden stop exceeds 10 percent of GDP.
  - Optimal level of reserves is zero if short-term debt amounts to less than 2.5 percent of GDP.
- Sensitivity to probability of sudden stop, term premium, and risk aversion:
  - Doubling probability of a sudden stop from 5 percent to 10 percent increases optimal reserves from 4.6 percent to 10.1 percent of GDP.
  - Increasing term premium from 1.5 percent to 3 percent reduces optimal reserve-to-GDP ratio from 10.2 percent to 3.8 percent.
  - Risk aversion effects:
    - Increase in risk-aversion from 1 to 4 raises optimal reserves from 3 percent to 14 percent of GDP.
    - Increase in risk-aversion from 6 to 8 has a much milder impact due to strong concavity.

### Estimating sudden-stop risk and regional calibration
- Empirical estimation:
  - Probit estimation of probability of sudden stops π using 34 middle income countries during 1975-2003; explanatory variables are averages of first and second lags.
  - Preferred specification (regression 3.1) finds π decreases with pre-crisis growth performance and increases with currency real appreciation, public debt-to-GDP, financial openness (absolute gross inflows as a share of GDP), and ratio of foreign liabilities to money in the banking sector.
  - Fixed exchange rate regime dummy associated with higher probability of sudden stop only if financial openness and exchange rate overvaluation are omitted.
- Size of sudden stops:
  - Unable to robustly estimate λ; take average size of sudden stop = 10.8 percent of GDP.
- Benchmark predicted probability and reserves:
  - Probit model (regression 3.1) predicts π = 7.9 percent for the average economy, implying optimal reserves-to-GDP = 8.6 percent.
- Impact of changes in fundamentals on optimal reserves (as reported in Table 5):
  - 20 percent real appreciation → estimated annual π rises by approximately 4.3 percentage points; optimal reserves increase by 2.7 percentage points of GDP.
  - Public debt-to-GDP rising from 40 percent to 60 percent → optimal reserve ratio increases by 1.7 percentage points of GDP.
  - Buildup of foreign liabilities to money (Thailand example: 1990 45 percent to 1997 262 percent) → optimal reserves raise by 3.7 percentage points of GDP.
  - One standard deviation increase in financial openness → optimal reserves increase by 3.2 percentage points of GDP.
  - Change from floating to fixed exchange rate regime → optimal reserves increase by 3.4 percentage points of GDP.

### Regional trends and model performance
- Aggregation approach:
  - Optimal reserves computed for each country-year for 34 middle-income countries; probability of sudden stop from probit estimates; size of sudden stop set to realized mean per region and decade; risk aversion set to 6 to match mean level of reserves-to-GDP in Asia in 1991.
- Asia:
  - Model suggests reserves should have declined between early and late 1980s due to slowdown in financial flows after early 1980s debt crisis.
  - Beginning early 1990s, model implies rapid increase in optimal reserves owing to rising international financial integration.
  - Slight decline in optimal reserves after Asian crises mainly due to reductions in public debt and financial flows.
  - Model outperforms three-months-of-imports and Greenspan-Guidotti rules in predicting actual reserve levels for most of the period considered.
  - Post-1997 buildup: model suggests buildup in reserves has been excessive. To justify observed accumulation between 1997 and 2003, expected size of either the sudden stop or the output cost would have had to more than double relative to its 1990s average. Increasing risk-aversion from 6 to 10 would raise warranted reserves by less than 3 percentage points of GDP—insufficient to explain the post-crisis buildup.
- Latin America:
  - Model suggests relatively high optimal reserves in 1980s, lower in early 1990s (improved fundamentals), and rising during the past decade with heightened financial integration.
  - Close match between model and actual data in 1991-2003 despite no individual-year Latin America reserves being used in calibration.
  - On average, current reserves may be adequate, though some large individual countries have insufficient reserve coverage.

### Extensions and robustness checks
- A. Crisis prevention (endogenous probability of sudden stop)
  - Model variant assumes π decreases with pre-crisis reserves Π(ρt) with Π′ < 0.
  - If sudden stop can be triggered by sunspot when short-term debt not fully covered by reserves:
    - Π(ρ) = sπ for ρ < λρs; Π(ρ) = 0 for ρ ≥ λρs (notation from text).
  - Government chooses between (1) setting ρλ = (Greenspan-Guidotti rule) or (2) setting ρ to optimal level conditional on π = sπ.
  - For benchmark calibration (Table 2) with sπ = 11 percent, Greenspan-Guidotti rule is optimal: increasing reserves from 92 percent to 100 percent of short-term debt is worth the cost.
  - Endogenizing π could increase or decrease optimal reserves in general.
- B. Real exchange rate depreciation
  - If external liabilities and reserves are in foreign currency and real exchange rate depreciates by ΔQ at sudden stop, the value of reserves in domestic consumption rises.
  - Ambiguous net effect: lower cost of insurance but same insurance achievable with fewer reserves.
  - For benchmark calibration, real depreciation of 10 percent increases optimal reserves from 10.1 percent to 10.2 percent of GDP (negligible change).
- C. Endogenizing private-sector behavior
  - Representative consumer faces private borrowing constraint b_t ≤ λY_t^b before sudden stop.
  - Constraint is binding if marginal utility condition holds; reformulated condition (19):
    - (1)1. 1
      g
      σ
      δ
      δπ
      +>+
      −−
  - Condition satisfied for benchmark calibration, so assumed private behavior consistent with intertemporal optimization.
  - Government reserves reduce private incentives to accumulate reserves; absence of government reserves makes private accumulation more likely, but restricting private short-term debt issuance would be welfare-decreasing in model.
- D. Multiple sudden stops
  - Sudden-stop episodes can last θ periods; full episode starting in t lasts until t+θ.
  - During episode, for s = 0,...,θ:
    - Y_{t+s} = Y_t^b − γ_s
    - L_{t+s} = λ_s L_t^b
  - Profile of episode determined by exogenous parameters {γ_s, λ_s} for s = 0,...,θ.

*Source: _wp06229 - 8.2 percent on average in our sample according to the World Bank’s Global Development (IMF working paper PDF content).*

### Section III corresponds to the special case where

### _wp06229 - Section III corresponds to the special case where

### Model special case and implication for reserves
- Section III corresponds to the special case where θ = +∞, 0γ >, 0s γ = for 0s >, and 0s λ = for all s.
- Assume the profile of the sudden-stop is such that it is optimal for the government to use all the reserves in the first period of the sudden-stop episode:
  - The government’s problem is exactly the same as in the model of Section III.
  - In normal times the government does not inherit any liability from the previous sudden-stop episode.
  - The government accumulates the optimal level of reserves to smooth domestic consumption in the first period of the next sudden-stop episode.
  - The formula for the optimal level of reserves derived earlier therefore remains valid under these assumptions.

### Appendix: computations (deriving key equations)
- Dynamics of output and private external debt summarized by:
  - Yt+1 = (1 + g)Yt and Yt+1^a = (1 + g)(1 − γ)Yt (notation in source: batdt ttt YYgYYgYγ ==+=− + etc.)
  - Lt+1^d = (1 + g)Lt and L^b dynamics with λ (source notation preserved).
- Substitution of these dynamics and bt−1Rt/ρ = Yt−1 into first-order conditions (equations (11) and (12) in source) yields expressions for C_bt+1 and C_dt+1 in terms of parameters g, λ, δ, π, ρ, γ.
- First-order condition (db)CpCσσ−− = g (source notation) combined with those expressions yields equation (16) in the paper.
- Case with a real exchange rate depreciation:
  - Government problem and constraints adjusted to include Q and ∆ terms (source equations preserved).
  - p is given by p = (1 − π)/(πδ + π) − +∆ (source notation).
  - Final manipulated expression (source) for p and reserves involves p, Q, r, g, λ, γ, ρ, δ, π, σ in the form shown in the source.

### Concluding comments — substantive findings and research directions
- Core contribution:
  - Derives a simple formula for the optimal level of international reserves based on the assumption that the main benefit of reserves is to smooth domestic absorption against disruptions induced by sudden stops in capital flows.
- Three directions for future work highlighted:
  1. Role of default risk in the opportunity cost of holding reserves:
     - Default risk premium is generally included in interest rate differentials used to compute the opportunity cost of holding reserves (Rodrik, 2006).
     - It is unclear whether the default risk premium—a compensation for risk that debt will not be repaid—should be counted like the term premium.
     - A default or suspension of debt service may itself be a disruptive smoothing device for domestic absorption; rigorous analysis of implications for optimal reserves requires further research.
  2. Preventive role of reserves in reducing probability and severity of sudden stops:
     - Extending the framework so that the probability of sudden stop—or other variables—are endogenous to reserve levels raises no fundamental conceptual problem.
     - Main difficulty is empirical estimation of the two-way causal relationship between reserves and these variables, necessary to calibrate the extended model.
  3. Collective management of reserves:
     - Investigate benefits of reserve pooling between emerging market countries.
     - Analyze consequences for reserve accumulation and domestic welfare of an institution such as the IMF that relaxes the external credit constraint of emerging market countries in a crisis.
     - These questions invite a multicountry extension of the framework.

### Empirical evidence and calibration (key parameters and regression findings)
- Table 2 — Calibration parameters (Baseline and ranges preserved exactly):
  - Size of sudden stop λ = 0.11 [0,0.3]
  - Probability of a sudden stop π = 0.10 [0,0.25]
  - Output loss γ = 0.065 [0,0.2]
  - Potential output growth g = 0.033
  - Term premium δ = 0.015 [0.0025,0.05]
  - Risk free rate r = 0.05
  - Risk aversion σ = 2 [1,10]
  - Source: Author’s calculations using data from IMF, International Financial Statistics, and the U.S. Federal Reserve Board.
- Table 3 — Probit estimation (selected coefficient signs and significance preserved):
  - Real effective exchange rate deviation from HP-trend: positive coefficients (e.g., 1.515 (3.12)*** in regression (3.1))
  - GDP growth: negative coefficients (e.g., −1.254 (1.62)*)
  - Public debt/GDP: positive and significant (e.g., 0.807 (3.33)***)
  - Ratio of foreign liabilities to money in the banking sector: positive and significant (e.g., 0.225 (3.03)***)
  - Financial openness (|Gross Inflows|)/GDP: positive and significant (e.g., 9.106 (5.68)***)
  - Dummy for fixed exchange rate regime: positive coefficient in one specification (0.29 (1.77)*)
  - Observations vary across specifications (e.g., 707, 690, 537)
  - Pseudo R2 reported (e.g., 0.14, 0.06, 0.17)
  - Time effects and fixed effects vary across regressions (as shown in source).
- Table 4 — Variables used in regression analysis (selected categories preserved):
  - Debt: Lag of real public debt to real GDP; Lag of short term debt to real GDP (GDF/WDI (2005))
  - Exchange rate: Second lag exchange rate regime dummies (Reinhart & Rogoff (2004)); Lag of real effective exchange rate deviation from HP trend (IFS (2005))
  - Financial account openness: Lag of absolute gross inflows/GDP (IFS (2005)), etc.
  - Databases: International Financial Statistics (IFS), Global Development Finance (GDF), World Development Indicators (WDI).
- Table 5 — Changes in fundamentals and impact on optimal reserves (values preserved exactly):
  - Initial optimal level of reserves: equivalent to 8.2 percent of GDP (about 3 months of imports, or 105 percent of short-term debt).
  - Exchange rate overvaluation: Sample mean 0.0; parameter change 0 → 20; estimated change in sudden-stop probability 4.3; change in optimal ratio of reserves to GDP 2.7.
  - Public debt/GDP: Sample mean 40.0; parameter change 40 → 60; estimated change in sudden-stop probability 2.2; change in optimal ratio of reserves to GDP 1.7.
  - Foreign liabilities / money: Sample mean 46.0; parameter change 45 (Thailand, 1990) → 262 (Thailand, 1997); estimated change in sudden-stop probability 7.0; change in optimal ratio of reserves to GDP 3.7.
  - Financial openness defined as (|gross inflows|)/GDP: Sample mean 5.5; parameter change 5.5 → 10.4; estimated change in sudden-stop probability 5.4; change in optimal ratio of reserves to GDP 3.2.
  - Exchange rate regime: Floating → Fixed; estimated change in sudden-stop probability 5.3; change in optimal ratio of reserves to GDP 3.4.
  - Note: Results based on regression (3.1) in Table 3—except for exchange rate regime, based on regression (3.2). All fundamentals are averages of the first and second lags.

### Figures and model calibration notes (selected)
- Figure 3 model calibration:
  - Reserves assumed to amount to 6 percent of GDP before the sudden stop.
  - Other parameters set to: 0.03, 0.1, 0.02, 0.1, 0.05, 0.015 with notation grλγπδ=== == = in source.
- Figure 4 shows sensitivity analysis for optimal reserves against:
  - Size of sudden stop (in percent of GDP)
  - Sudden stop probability (in percent)
  - Opportunity cost of holding reserves (in basis points)
  - Risk aversion
  - Comparison lines include Greenspan-Guidotti Rule and Jeanne-Rancière Model (as in source).
- Figure 1 and Figure 5 present cross-country and regional reserves as share of GDP time series (data sources: IMF IFS, WEO, World Bank WDI; Figure notes preserved in source).

*Source: _wp06229 - Section III corresponds to the special case where (IMF PDF content provided).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp06229.pdf_